16 research outputs found
Fast, high-order solution of surface scattering problems
We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates the scattered field, through fast, high-order solution of the boundary integral equation. The high-order of the solver is achieved through use of partition of unity together with analytical resolution of kernel singularities. The acceleration in turn, results from a novel approach which, based on high-order "two-face" equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D FFTs. We demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer, scattering from bodies of acoustical sizes of several hundreds
A mathematical model and inversion procedure for Magneto-Acousto-Electric Tomography (MAET)
Magneto-Acousto-Electric Tomography (MAET), also known as the Lorentz force
or Hall effect tomography, is a novel hybrid modality designed to be a
high-resolution alternative to the unstable Electrical Impedance Tomography. In
the present paper we analyze existing mathematical models of this method, and
propose a general procedure for solving the inverse problem associated with
MAET. It consists in applying to the data one of the algorithms of
Thermo-Acoustic tomography, followed by solving the Neumann problem for the
Laplace equation and the Poisson equation.
For the particular case when the region of interest is a cube, we present an
explicit series solution resulting in a fast reconstruction algorithm. As we
show, both analytically and numerically, MAET is a stable technique yilelding
high-resolution images even in the presence of significant noise in the data
2D and 3D reconstructions in acousto-electric tomography
We propose and test stable algorithms for the reconstruction of the internal
conductivity of a biological object using acousto-electric measurements.
Namely, the conventional impedance tomography scheme is supplemented by
scanning the object with acoustic waves that slightly perturb the conductivity
and cause the change in the electric potential measured on the boundary of the
object. These perturbations of the potential are then used as the data for the
reconstruction of the conductivity. The present method does not rely on
"perfectly focused" acoustic beams. Instead, more realistic propagating
spherical fronts are utilized, and then the measurements that would correspond
to perfect focusing are synthesized. In other words, we use \emph{synthetic
focusing}. Numerical experiments with simulated data show that our techniques
produce high quality images, both in 2D and 3D, and that they remain accurate
in the presence of high-level noise in the data. Local uniqueness and stability
for the problem also hold
Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm
Practical applications of thermoacoustic tomography require numerical
inversion of the spherical mean Radon transform with the centers of integration
spheres occupying an open surface. Solution of this problem is needed (both in
2-D and 3-D) because frequently the region of interest cannot be completely
surrounded by the detectors, as it happens, for example, in breast imaging. We
present an efficient numerical algorithm for solving this problem in 2-D
(similar methods are applicable in the 3-D case). Our method is based on the
numerical approximation of plane waves by certain single layer potentials
related to the acquisition geometry. After the densities of these potentials
have been precomputed, each subsequent image reconstruction has the complexity
of the regular filtration backprojection algorithm for the classical Radon
transform. The peformance of the method is demonstrated in several numerical
examples: one can see that the algorithm produces very accurate reconstructions
if the data are accurate and sufficiently well sampled, on the other hand, it
is sufficiently stable with respect to noise in the data
Fast, high-order solution of surface scattering problems
We present a fast, high-order algorithm for the solution of problems of acoustic scattering from smooth surfaces in three dimensions. The present algorithm computes scattered fields i
Fast, high-order solution of surface scattering problems
We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates the scattered field, through fast, high-order solution of the boundary integral equation. The high-order of the solver is achieved through use of partition of unity together with analytical resolution of kernel singularities. The acceleration in turn, results from a novel approach which, based on high-order "two-face" equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D FFTs. We demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer, scattering from bodies of acoustical sizes of several hundreds
A series solution and a fast algorithm for the inversion of the spherical mean Radon transform
An explicit series solution is proposed for the inversion of the spherical
mean Radon transform. Such an inversion is required in problems of thermo- and
photo- acoustic tomography. Closed-form inversion formulae are currently known
only for the case when the centers of the integration spheres lie on a sphere
surrounding the support of the unknown function, or on certain unbounded
surfaces. Our approach results in an explicit series solution for any closed
measuring surface surrounding a region for which the eigenfunctions of the
Dirichlet Laplacian are explicitly known - such as, for example, cube, finite
cylinder, half-sphere etc. In addition, we present a fast reconstruction
algorithm applicable in the case when the detectors (the centers of the
integration spheres) lie on a surface of a cube. This algorithm reconsrtucts
3-D images thousands times faster than backprojection-type methods
Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra
We present explicit filtration/backprojection-type formulae for the inversion
of the spherical (circular) mean transform with the centers lying on the
boundary of some polyhedra (or polygons, in 2D). The formulae are derived using
the double layer potentials for the wave equation, for the domains with certain
symmetries. The formulae are valid for a rectangle and certain triangles in 2D,
and for a cuboid, certain right prisms and a certain pyramid in 3D. All the
present inversion formulae yield exact reconstruction within the domain
surrounded by the acquisition surface even in the presence of exterior sources.Comment: 9 figure
Spectral Properties of High Contrast Band-Gap Materials and Operators on Graphs
The theory of classical waves in periodic high contrast photonic and acoustic media leads to the spectral problem
- Îu= λâu, where the dielectric constant â(x) is a periodic function which assumes a large value â near a periodic graph ÎŁ in R^2 and is equal to 1 otherwise. Existence and locations of spectral gaps are of primary interest. The high contrast asymptotics naturally
leads to pseudodifferential operators of the Dirichlet-to-Neumann type on graphs and on more general structures. Spectra of these operators are studied numerically and analytically. New spectral effects are discovered, among them the âalmost discretenessâ of the spectrum for a disconnected graph and the existence of âalmost localizedâ waves in some connected purely periodic structures